Properties

Label 4010.2.a.i
Level 4010
Weight 2
Character orbit 4010.a
Self dual Yes
Analytic conductor 32.020
Analytic rank 1
Dimension 10
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4010.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0200112105\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} -\beta_{7} q^{3} + q^{4} + q^{5} + \beta_{7} q^{6} + ( -1 + \beta_{3} + \beta_{7} ) q^{7} - q^{8} + ( \beta_{1} + \beta_{6} ) q^{9} +O(q^{10})\) \( q - q^{2} -\beta_{7} q^{3} + q^{4} + q^{5} + \beta_{7} q^{6} + ( -1 + \beta_{3} + \beta_{7} ) q^{7} - q^{8} + ( \beta_{1} + \beta_{6} ) q^{9} - q^{10} + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{11} -\beta_{7} q^{12} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( 1 - \beta_{3} - \beta_{7} ) q^{14} -\beta_{7} q^{15} + q^{16} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{17} + ( -\beta_{1} - \beta_{6} ) q^{18} + ( -1 - \beta_{5} - \beta_{7} - \beta_{8} ) q^{19} + q^{20} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{21} + ( 1 + \beta_{6} - \beta_{7} - \beta_{9} ) q^{22} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{23} + \beta_{7} q^{24} + q^{25} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{26} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{27} + ( -1 + \beta_{3} + \beta_{7} ) q^{28} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{29} + \beta_{7} q^{30} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{31} - q^{32} + ( -2 + 2 \beta_{4} + 3 \beta_{7} ) q^{33} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} ) q^{34} + ( -1 + \beta_{3} + \beta_{7} ) q^{35} + ( \beta_{1} + \beta_{6} ) q^{36} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{37} + ( 1 + \beta_{5} + \beta_{7} + \beta_{8} ) q^{38} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} + \beta_{8} ) q^{39} - q^{40} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{41} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{42} + ( 2 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{43} + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{44} + ( \beta_{1} + \beta_{6} ) q^{45} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{46} + ( -1 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 3 \beta_{8} ) q^{47} -\beta_{7} q^{48} + ( -2 \beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{49} - q^{50} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{51} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{52} + ( 4 - 2 \beta_{1} - \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{53} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{54} + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{55} + ( 1 - \beta_{3} - \beta_{7} ) q^{56} + ( 3 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{57} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{9} ) q^{58} + ( -4 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{59} -\beta_{7} q^{60} + ( -5 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{61} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{9} ) q^{62} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{63} + q^{64} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{65} + ( 2 - 2 \beta_{4} - 3 \beta_{7} ) q^{66} + ( -2 + \beta_{4} + \beta_{8} + 2 \beta_{9} ) q^{67} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{68} + ( -2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{8} ) q^{69} + ( 1 - \beta_{3} - \beta_{7} ) q^{70} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{71} + ( -\beta_{1} - \beta_{6} ) q^{72} + ( -2 + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{7} ) q^{73} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} ) q^{74} -\beta_{7} q^{75} + ( -1 - \beta_{5} - \beta_{7} - \beta_{8} ) q^{76} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{77} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} - \beta_{8} ) q^{78} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{79} + q^{80} + ( -2 + 3 \beta_{1} + \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{81} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{82} + ( -\beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} ) q^{83} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{84} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{85} + ( -2 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} ) q^{86} + ( 5 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{87} + ( 1 + \beta_{6} - \beta_{7} - \beta_{9} ) q^{88} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{89} + ( -\beta_{1} - \beta_{6} ) q^{90} + ( -5 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{91} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{92} + ( -4 + 3 \beta_{1} + 6 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} ) q^{93} + ( 1 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + 3 \beta_{8} ) q^{94} + ( -1 - \beta_{5} - \beta_{7} - \beta_{8} ) q^{95} + \beta_{7} q^{96} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} + 3 \beta_{8} - 2 \beta_{9} ) q^{97} + ( 2 \beta_{2} - \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{98} + ( -4 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 10q^{2} - 4q^{3} + 10q^{4} + 10q^{5} + 4q^{6} - 3q^{7} - 10q^{8} + 6q^{9} + O(q^{10}) \) \( 10q - 10q^{2} - 4q^{3} + 10q^{4} + 10q^{5} + 4q^{6} - 3q^{7} - 10q^{8} + 6q^{9} - 10q^{10} - 11q^{11} - 4q^{12} + 6q^{13} + 3q^{14} - 4q^{15} + 10q^{16} + 9q^{17} - 6q^{18} - 13q^{19} + 10q^{20} - 24q^{21} + 11q^{22} - 3q^{23} + 4q^{24} + 10q^{25} - 6q^{26} - 10q^{27} - 3q^{28} - 4q^{29} + 4q^{30} - 17q^{31} - 10q^{32} - 2q^{33} - 9q^{34} - 3q^{35} + 6q^{36} - 15q^{37} + 13q^{38} - 6q^{39} - 10q^{40} - 11q^{41} + 24q^{42} - 11q^{43} - 11q^{44} + 6q^{45} + 3q^{46} + 3q^{47} - 4q^{48} - 5q^{49} - 10q^{50} - 21q^{51} + 6q^{52} + 25q^{53} + 10q^{54} - 11q^{55} + 3q^{56} + 31q^{57} + 4q^{58} - 46q^{59} - 4q^{60} - 54q^{61} + 17q^{62} - 6q^{63} + 10q^{64} + 6q^{65} + 2q^{66} - 26q^{67} + 9q^{68} - 9q^{69} + 3q^{70} - 16q^{71} - 6q^{72} + 4q^{73} + 15q^{74} - 4q^{75} - 13q^{76} + 11q^{77} + 6q^{78} - 19q^{79} + 10q^{80} - 6q^{81} + 11q^{82} + 19q^{83} - 24q^{84} + 9q^{85} + 11q^{86} + 28q^{87} + 11q^{88} - 30q^{89} - 6q^{90} - 38q^{91} - 3q^{92} - 18q^{93} - 3q^{94} - 13q^{95} + 4q^{96} - 16q^{97} + 5q^{98} - 59q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 3 x^{9} - 12 x^{8} + 34 x^{7} + 46 x^{6} - 104 x^{5} - 90 x^{4} + 89 x^{3} + 82 x^{2} + 12 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -129 \nu^{9} + 529 \nu^{8} + 1066 \nu^{7} - 5745 \nu^{6} - 889 \nu^{5} + 16145 \nu^{4} - 1628 \nu^{3} - 12944 \nu^{2} - 352 \nu - 343 \)\()/647\)
\(\beta_{3}\)\(=\)\((\)\( 388 \nu^{9} - 1235 \nu^{8} - 4415 \nu^{7} + 14195 \nu^{6} + 14355 \nu^{5} - 45275 \nu^{4} - 18596 \nu^{3} + 45939 \nu^{2} + 6646 \nu - 5975 \)\()/1941\)
\(\beta_{4}\)\(=\)\((\)\( -623 \nu^{9} + 1978 \nu^{8} + 7024 \nu^{7} - 21847 \nu^{6} - 24225 \nu^{5} + 63547 \nu^{4} + 45307 \nu^{3} - 52647 \nu^{2} - 39236 \nu - 1817 \)\()/1941\)
\(\beta_{5}\)\(=\)\((\)\( -304 \nu^{9} + 1041 \nu^{8} + 3119 \nu^{7} - 11402 \nu^{6} - 8239 \nu^{5} + 32505 \nu^{4} + 11215 \nu^{3} - 26075 \nu^{2} - 11337 \nu - 61 \)\()/647\)
\(\beta_{6}\)\(=\)\((\)\( -913 \nu^{9} + 2771 \nu^{8} + 10574 \nu^{7} - 31166 \nu^{6} - 36405 \nu^{5} + 94355 \nu^{4} + 57125 \nu^{3} - 83391 \nu^{2} - 41542 \nu + 998 \)\()/1941\)
\(\beta_{7}\)\(=\)\((\)\( 983 \nu^{9} - 3364 \nu^{8} - 10360 \nu^{7} + 37699 \nu^{6} + 29640 \nu^{5} - 113839 \nu^{4} - 45052 \nu^{3} + 103395 \nu^{2} + 46583 \nu - 205 \)\()/1941\)
\(\beta_{8}\)\(=\)\((\)\( -1162 \nu^{9} + 4409 \nu^{8} + 10811 \nu^{7} - 48665 \nu^{6} - 19689 \nu^{5} + 142145 \nu^{4} + 10769 \nu^{3} - 125544 \nu^{2} - 20404 \nu + 7799 \)\()/1941\)
\(\beta_{9}\)\(=\)\((\)\( -1720 \nu^{9} + 5975 \nu^{8} + 18311 \nu^{7} - 68189 \nu^{6} - 52830 \nu^{5} + 213110 \nu^{4} + 74912 \nu^{3} - 201486 \nu^{2} - 72844 \nu + 4916 \)\()/1941\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{8} + \beta_{6} - \beta_{5} + 2 \beta_{3} - \beta_{2} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + 3 \beta_{8} - \beta_{7} + 10 \beta_{6} - 12 \beta_{5} + \beta_{4} + 10 \beta_{3} + 6 \beta_{2} + 11 \beta_{1} + 19\)
\(\nu^{5}\)\(=\)\(13 \beta_{8} + 3 \beta_{7} + 15 \beta_{6} - 15 \beta_{5} + \beta_{4} + 24 \beta_{3} - 9 \beta_{2} + 56 \beta_{1} + 9\)
\(\nu^{6}\)\(=\)\(-10 \beta_{9} + 41 \beta_{8} - 2 \beta_{7} + 93 \beta_{6} - 114 \beta_{5} + 16 \beta_{4} + 97 \beta_{3} + 37 \beta_{2} + 108 \beta_{1} + 135\)
\(\nu^{7}\)\(=\)\(\beta_{9} + 142 \beta_{8} + 52 \beta_{7} + 171 \beta_{6} - 180 \beta_{5} + 29 \beta_{4} + 250 \beta_{3} - 74 \beta_{2} + 478 \beta_{1} + 85\)
\(\nu^{8}\)\(=\)\(-74 \beta_{9} + 449 \beta_{8} + 70 \beta_{7} + 848 \beta_{6} - 1042 \beta_{5} + 195 \beta_{4} + 927 \beta_{3} + 229 \beta_{2} + 1046 \beta_{1} + 1014\)
\(\nu^{9}\)\(=\)\(25 \beta_{9} + 1462 \beta_{8} + 660 \beta_{7} + 1784 \beta_{6} - 1969 \beta_{5} + 445 \beta_{4} + 2508 \beta_{3} - 600 \beta_{2} + 4229 \beta_{1} + 848\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.08731
1.47143
−1.54628
0.0585304
−0.557507
1.96171
−2.65028
−1.06458
−0.309245
2.54891
−1.00000 −3.13000 1.00000 1.00000 3.13000 2.49851 −1.00000 6.79692 −1.00000
1.2 −1.00000 −2.70072 1.00000 1.00000 2.70072 −1.21166 −1.00000 4.29386 −1.00000
1.3 −1.00000 −1.74362 1.00000 1.00000 1.74362 2.79885 −1.00000 0.0402221 −1.00000
1.4 −1.00000 −1.47624 1.00000 1.00000 1.47624 −2.32277 −1.00000 −0.820714 −1.00000
1.5 −1.00000 −1.25862 1.00000 1.00000 1.25862 1.88678 −1.00000 −1.41588 −1.00000
1.6 −1.00000 0.223737 1.00000 1.00000 −0.223737 −0.465545 −1.00000 −2.94994 −1.00000
1.7 −1.00000 0.675556 1.00000 1.00000 −0.675556 −1.96617 −1.00000 −2.54362 −1.00000
1.8 −1.00000 0.696154 1.00000 1.00000 −0.696154 2.65990 −1.00000 −2.51537 −1.00000
1.9 −1.00000 2.30797 1.00000 1.00000 −2.30797 −5.12569 −1.00000 2.32671 −1.00000
1.10 −1.00000 2.40579 1.00000 1.00000 −2.40579 −1.75221 −1.00000 2.78781 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(401\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\(T_{3}^{10} + \cdots\)
\(T_{7}^{10} + \cdots\)
\(T_{11}^{10} + \cdots\)